26 Combinatorial Analysis Properties 26.16 Multiset Permutations 26.18 Counting Techniques

§26.17 The Twelvefold Way

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Keywords:: permutations, twelvefold way
Permalink:: http://dlmf.nist.gov/26.17
See also:: Annotations for Ch.26

The twelvefold way gives the number of mappings $f$ from set $N$ of $n$ objects to set $K$ of $k$ objects (putting balls from set $N$ into boxes in set $K$ ). See Table 26.17.1. In this table ${\left(k\right)_{n}}$ is Pochhammer’s symbol, and $S\left(n,k\right)$ and $p_{k}\left(n\right)$ are defined in §§26.8(i) and 26.9(i).

Table 26.17.1 is reproduced (in modified form) from Stanley (1997, p. 33). See also Example 3 in §26.18.

Table 26.17.1: The twelvefold way.

elements of $N$	elements of $K$	$f$ unrestricted	$f$ one-to-one	$f$ onto
labeled	labeled	$k^{n}$	${\left(k-n+1\right)_{n}}$	$k!S\left(n,k\right)$
unlabeled	labeled	$\dbinom{k+n-1}{n}$	$\dbinom{k}{n}$	$\dbinom{n-1}{n-k}$
labeled	unlabeled	$\begin{array}[]{c}S\left(n,1\right)+S\left(n,2\right)\\ +\cdots+S\left(n,k\right)\end{array}$	$\begin{cases}1&n\leq k\\ 0&n>k\end{cases}$	$S\left(n,k\right)$
unlabeled	unlabeled	$p_{k}\left(n\right)$	$\begin{cases}1&n\leq k\\ 0&n>k\end{cases}$	$p_{k}\left(n\right)-p_{k-1}\left(n\right)$

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $p_{\NVar{k}}\left(\NVar{n}\right)$ : total number of partitions of $n$ into at most $k$ parts, $k$ : nonnegative integer, $n$ : nonnegative integer, $f$ : mapping and $N$ : set
Referenced by:: §26.17, §26.17, §26.18
Permalink:: http://dlmf.nist.gov/26.17.T1
See also:: Annotations for §26.17 and Ch.26

26.16 Multiset Permutations 26.18 Counting Techniques

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§26.17 The Twelvefold Way

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